Faá di Bruno's Formula

Author

Eric W. Weisstein
April 14, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/FaadiBrunosFormula.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FaadiBrunosFormula.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<DiscreteMath`Combinatorica`

Code

PartitionsJ[p_]:=Reduce`NaturalLinearSolve[{Range[p]},{p},True][[1]]

FaaDiBruno[f_,g_,t_,n_]:=n! Plus@@(Derivative[Plus@@#][f][g[t]]/Times@@Factorial[#]Times@@MapIndexed[(Derivative[#2[[1]]][g][t]/#2[[1]]!)^#1&,#]&/@PartitionsJ[n])

Examples

Simplify[Table[FaaDiBruno[f,g,t,n],{n,3}]]//ColumnForm

Check

Simplify[Table[D[f[g[t]],{t,n}]-Simplify[FaaDiBruno[f,g,t,n]],{n,10}]]